Need to show $K(\alpha,\beta)$ is not a simple extension of $K(\alpha^p,\beta^p)$

Question

Let $K$ be a field of characteristic $p\not=0$, and $K(\alpha)$ and $K(\beta)$ are both transcendental simple extensions of $K$ such that $\alpha \notin K(\beta)$ and $\beta \notin K(\alpha)$.

I need help to show that $K(\alpha,\beta)$ is not a simple extension of $K(\alpha^p,\beta^p)$.

Answer 1

Hint

Prove that $[K(\alpha,\beta):K(\alpha^p,\beta^p)] = p^2$ by first proving that $[K(\alpha) : K(\alpha^p)] = p$.

Then notice that for any $\gamma \in K(\alpha,\beta)$ we have $[K(\alpha^p,\beta^p)(\gamma) : K(\alpha^p,\beta^p)] \le p$.